Showing papers on "Fourier series published in 2019"

Free vibrations of functionally graded porous rectangular plate with uniform elastic boundary conditions

Abstract: In this paper, the free vibrations of functionally graded porous (FGP) rectangular plate with uniform elastic boundary conditions is investigated by means of an improved Fourier series method (IFSM). It is assumed that the distributions of porosity are uniform or non-uniformly along a certain direction and three types of the porosity distribution are considered, among which material property of two non-uniform porous distributions was expressed as the simple cosine. The size of the pore in a rectangular plate is determined by the porosity coefficients. Using the first-order shear deformation theory(FSDT), the energy expression of FGP rectangular plate is created. In order to obtain the admissible function of displacement for functionally graded porous rectangular plate, the IFSM is employed. Then, the Rayleigh-Ritz method is used to solve coefficients in the Fourier series which determine natural frequencies and modal shapes. Convergence and comparative research are performed to prove the convergence, reliability and accuracy of the current method. On this foundation, some new results covering the influence of the geometrical parameters subject to classical and elastic boundary condition are presented, and the parametric studies are also investigated in detail, which can provide a reference for future research by other researchers.

A unified solution for the vibration analysis of functionally graded porous (FGP) shallow shells with general boundary conditions

Abstract: The main purpose of this paper is to illustrate the vibration characteristics of functionally graded porous (FGP) shallow shells with general boundary conditions for the first time The general boundary condition of FGP shallow shells is realized by the virtual spring technique The imposing procedures of the boundary conditions are simplified so that a certain kind of restraints can be easily achieved by merely setting different stiffness of the springs It is assumed that the distributions of porosity are uniform or non-uniformly along a certain direction and three types of the porosity distribution are considered, among which material property of two non-uniform porous distributions are expressed as the simple cosine The size of the pore in a shallow shell is determined by the porosity coefficients Based on the first-order shear deformation theory (FSDT), all kinetic energy and potential energy of FGP shallow shells are expressed by displacement admissible function On this basis, the author describes the displacement admissible function of the FGP shallow shells by using the modified Fourier series which increases the auxiliary function, so that the auxiliary function can be used to eliminate the discontinuity or jumping of the traditional Fourier series at the edges Lastly, the natural frequencies as well as the associated mode shapes of FGP shallow shells are achieved by replacing the modified Fourier series into the above energy expression and using the variational operation for unknown expansion coefficients Several numerical examples are carried out to demonstrate the validity and accuracy of the present solution by comparing with the results obtained by other researchers In addition, a series of innovative results are also highlighted in the text, which may provide basic data for other algorithm research in the future

Vibration behavior of the functionally graded porous (FGP) doubly-curved panels and shells of revolution by using a semi-analytical method

Abstract: The main goal of this article is to provide parameterization study for vibration behavior of functionally graded porous (FGP) doubly-curved panels and shells of revolution by using a semi-analytical method. Distribution of the porous through the thickness of structure may be uniform or non-uniform and three types of the porosity distributions are performed in this paper. Mechanical properties of materials are determined by open-cell metal foam. Energy expressions, including kinetic energy and potential energy, are expressed by displacement admissible function. Then, in order to obtain the general boundary conditions including the simply classical boundary conditions, elastic boundary constraint and their combinatorial boundary constraints, each of displacement admissible functions is expanded as a modified Fourier series of a standard cosine Fourier series with the auxiliary functions introduced to eliminate all potential discontinuities of the original displacement function and its derivatives at the edges. Lastly, the natural frequencies as well as the associated mode shapes of FGP doubly-curved panels and shells of revolution are achieved by replacing the modified Fourier series into the above energy expression and using the variational operation for unknown expansion coefficients. The convergence and accuracy of the present modeling are validated by comparing its results with those available in the literature and FEM results. Based on that, a series of innovative results are also highlighted in the text, which may provide basic data for other algorithm research in the future.

Jacobi–Ritz method for free vibration analysis of uniform and stepped circular cylindrical shells with arbitrary boundary conditions: A unified formulation

Abstract: A semi analytical approach is employed to analyze free vibration characteristics of uniform and stepped circular cylindrical shells subject to arbitrary boundary conditions. The analytical model is established on the basis of multi-segment partitioning strategy and Flugge thin shell theory. The admissible displacement functions are handled by unified Jacobi polynomials and Fourier series. In order to obtain continuous conditions and satisfy arbitrary boundary conditions, the penalty method about the spring technique is adopted. The solutions about free vibration behavior of circular cylindrical shells were obtained by approach of Rayleigh–Ritz. To confirm the reliability and accuracy of this method, convergence study and numerical verifications for circular cylindrical shells subject to different boundary conditions, Jacobi parameters, spring parameters and maximum degree of permissible displacement function are carried out. Through comparative analyses, it is obvious that the present method has a good stable and rapid convergence property and the results of this paper agree closely with the published literature. In addition, some interesting results about the geometric dimensions are investigated.

Application of flügge thin shell theory to the solution of free vibration behaviors for spherical-cylindrical-spherical shell: A unified formulation

Abstract: The main purpose of this paper is to provide a semi analytical method to analyze the free vibration of spherical-cylindrical-spherical shell subject to arbitrary boundary conditions. The formulations are established based on energy method and Flugge thin shell theory. The displacement functions are expressed by unified Jacobi polynomials and Fourier series . The arbitrary boundary conditions are simulated by penalty method about spring stiffness . The final solutions of spherical-cylindrical-spherical shell are obtained by Rayleigh–Ritz method. To sufficient illustrate the effectiveness of proposed method, some numerical example about spring stiffness, Jacobi parameters etc. are carried out. In addition, to verify the accuracy of this method, the results are compared with those obtained by FEM, experiment and published literature. The results show that the proposed method has ability to solve the free vibration behaviors of spherical-cylindrical-spherical shell.

Dynamics analysis of functionally graded porous (FGP) circular, annular and sector plates with general elastic restraints

Abstract: In this paper, the dynamics analysis of functionally graded porous (FGP) circular, annular and sector plates with general elastic restraints is performed in a unified form for the first time. The overall theoretical model is based on the first order shear deformation theory. The kinetic energy and potential energy function of the plates are unified representation of five kinds of displacement admissible function. Then, each of displacement admissible function is expanded as a modified Fourier series to obtain general elastic restraints. Lastly, the solutions are obtained by using the variational operation. The convergence and accuracy of the present modeling are validated by comparing its results with those available in the literature and FEM results. Based on that, a series of innovative results are also highlighted in the text, which may be as the basic data for other algorithm research in the future.

Free vibration characteristics of functionally graded porous spherical shell with general boundary conditions by using first-order shear deformation theory

Abstract: The paper analyzed the free vibration of functionally graded porous spherical shell (FGPSS) based on Ritz method. The energy method and first-order shear deformation theory (FSDT) are adopted to derive the formulas. In this paper, the displacement functions are improved on basis of domain decomposition method, in which the unified Jacobi polynomials are introduced to represent the displacement functions component along meridional direction, and the displacement functions component along circumferential direction is still Fourier series. In addition, the spring stiffness method is formed a unified format to deal with various complex boundary conditions and continuity conditions. Then the final solutions can be obtained based on Ritz method. To prove the validity of proposed method, the results of the same condition are compared with those obtained by FEM, published literatures and experiment. The results show that the proposed method has advantages of fast convergence, high calculation efficiency, high solution accuracy and simple boundary simulation.

A modified series solution for free vibration analyses of moderately thick functionally graded porous (FGP) deep curved and straight beams

Abstract: As a novel class of weight-efficient engineering materials, the functionally graded porous (FGP) beam structures have great potential value. However, the current research on it is relatively small. Based on this research status, the aim of this paper is establishing a unified analytical model to study the vibration behavior of moderately thick functionally graded porous deep curved and straight beam with general boundary conditions. The first-order beam theory which considering the influence of shear deformation, inertia rotary and deepness term are adopted in the formulation. The theoretical solution model is obtained by means of modified series solution which core soul is using the modified Fourier series including a standard cosine Fourier series with two auxiliary terms to expand the admissible function. This fact gives the opportunity to derive the exact solution for FGP beam with general boundary conditions by utilizing a reasonable spring stiffness value at both ends. A series of numerical examples show that the current model has superior convergence characteristics, computational accuracy and stability. On this basis, a series of innovative results are also highlighted in the text, which may be providing basic data for other algorithm research in the future.

Transient response in a thermoelastic half-space solid due to a laser pulse under three theories with memory-dependent derivative

Abstract: Enlightened by the Caputo fractional derivative, the present study deals with a novel mathematical model of generalized thermoelasticity to investigate the transient phenomena due to the influence of a non-Gaussian pulsed laser type heat source in a stress free isothermal half-space in the context of Lord–Shulman (LS), dual-phase lag (DPL), and three-phase lag (TPL) theories of thermoelasticity simultaneously. The memory-dependent derivative is defined in an integral form of a common derivative on a slipping interval by incorporating the memory-dependent heat transfer. Employing Laplace transform as a tool, the problem has been transformed to the space-domain, and it is then solved analytically. To get back all the thermophysical quantities as a function of real time, we use two Laplace inversion formulas, viz. Fourier series expansion technique (Honig in J Comput Appl Math10(1):113–132, 1984) and Zakian method (Electron Lett 6(21):677–679, 1970). According to the graphical representations corresponding to the numerical results, a comparison among LS, DPL, and TPL model has been studied in the presence and absence of a memory effect simultaneously. Moreover, the effects of a laser pulse have been studied in all the thermophysical quantities for different kernels (randomly chosen) and different delay times. Then, the results are depicted graphically. Finally, a comparison of results, deriving from the two numerical inversion formulas, has been made.

Product decomposition of periodic functions in quantum signal processing

Abstract: In some embodiments, one or more unitary-valued functions are generated by a classical computer generating using projectors with a predetermined number of significant bits. A quantum computing device is then configured to implement the one or more unitary-valued functions. In further embodiments, a quantum circuit description for implementing quantum signal processing that decomposes complex-valued periodic functions is generated by a classical computer, wherein the generating further includes representing approximate polynomials in a Fourier series with rational coefficients. A quantum computing device is then configured to implement a quantum circuit defined by the quantum circuit description.

Transient response in a piezoelastic medium due to the influence of magnetic field with memory-dependent derivative

Abstract: Enlightened by the Caputo fractional derivative, the present study deals with a novel mathematical model of generalized thermoelasticity to investigate the transient phenomena for a piezoelastic half-space due to the influence of a magnetic field in the context of the dual-phase-lag model of generalized thermoelasticity, which is defined in an integral form of a common derivative on a slipping interval by incorporating the memory-dependent heat transfer. The bounding plane of the medium is assumed to be stress free and subjected to a thermal shock. Employing the Laplace transform as a tool, the problem is transformed to the space domain, where the solution in the space–time domain is achieved by applying a suitable numerical technique based on Fourier series expansion technique. According to the graphical representations corresponding to the numerical results, conclusions about the new theory are constructed. Excellent predictive capability is demonstrated due to the presence of electric field, memory-dependent derivative and magnetic field.

Onset of the wave turbulence description of the longtime behavior of the nonlinear Schr\"odinger equation

Abstract: Consider the cubic nonlinear Schrodinger equation set on a d-dimensional torus, with data whose Fourier coefficients have phases which are uniformly distributed and independent. We show that, on average, the evolution of the moduli of the Fourier coefficients is governed by the so-called wave kinetic equation, predicted in wave turbulence theory, on a nontrivial timescale.

Expansions of multiple Stratonovich stochastic integrals, based on generalized multiple Fourier series

Abstract: The article is devoted to the expansions of multiple Stratonovich stochastic integrals of multiplicities 1-4 on the basis of the method of generalized multiple Fourier series. Mean-square convergence of the expansions for the case of Legendre polynomials and for the case of trigonometric functions is proven. The considered expansions contain only one operation of the limit transition in contrast to its existing analogues. This property is comfortable for the mean-square approximation of multiple stochastic integrals. The results of the article can be applied to numerical integration of Ito stochastic differential equations.

Application of the Method of Approximation of Iterated Ito Stochastic Integrals Based on Generalized Multiple Fourier Series to the High-Order Strong Numerical Methods for Non-Commutative Semilinear Stochastic Partial Differential Equations.

Abstract: We consider a method for the approximation of iterated stochastic integrals of arbitrary multiplicity $k$ $(k\in \mathbb{N})$ with respect to the infinite-dimensional $Q$-Wiener process using the mean-square approximation method of iterated Ito stochastic integrals with respect to the scalar standard Wiener processes based on generalized multiple Fourier series. The case of multiple Fourier-Legendre series is considered in details. The results of the article can be applied to construction of high-order strong numerical methods (with respect to the temporal discretization) for the approximation of mild solution for non-commutative semilinear stochastic partial differential equations.

Modular graph functions and odd cuspidal functions -- Fourier and Poincar\'e series

Abstract: Modular graph functions are $SL(2,{\mathbb Z})$-invariant functions associated with Feynman graphs of a two-dimensional conformal field theory on a torus of modulus $\tau$. For one-loop graphs they reduce to real analytic Eisenstein series. We obtain the Fourier series, including the constant and non-constant Fourier modes, of all two-loop modular graph functions, as well as their Poincare series with respect to $\Gamma_\infty \backslash PSL(2,{\mathbb Z})$. The Fourier and Poincare series provide the tools to compute the Petersson inner product of two-loop modular graph functions using Rankin-Selberg-Zagier methods. Modular graph functions which are odd under $\tau \to - \bar \tau$ are cuspidal functions, with exponential decay near the cusp, and exist starting at two loops. Holomorphic subgraph reduction and the sieve algorithm, developed in earlier work, are used to give a lower bound on the dimension of the space $\mathfrak{A}_w$ of odd two-loop modular graph functions of weight $w$. For $w \leq 11$ the bound is saturated and we exhibit a basis for $\mathfrak{A}_w$.

A comparative example of cyclostationary description of a non-stationary random process

Abstract: The paper deals with cyclostationarity as a natural extension of stationarity as the key property in designing the widely-used models of random processes. The comparative example of two processes, one is wide-sense stationary and the other is wide-sense cyclostationary, is given in the paper and reveals the lack of the conventional stationary description based on one-dimensional autocorrelation functions. It is shown that two significantly different random processes appear to be characterized by exactly the same autocorrelation function while their two-dimensional autocorrelation functions provide outlook where the difference between processes of two above-mentioned classes becomes much clearer. More concise representation by expanding the two-dimensional autocorrelation function to its Fourier series where the cyclic frequency appears as the transform parameter is illustrated. The closed-form expression for the components of the cyclic autocorrelation function is also given for the random process which is an infinite pulse train made of rectangular pulses with randomly varying amplitudes.

A Comparative Analysis of Efficiency of Using the Legendre Polynomials and Trigonometric Functions for the Numerical Solution of Ito Stochastic Differential Equations

Abstract: This paper is devoted to the comparative analysis of the efficiency of using the Legendre polynomials and trigonometric functions for the numerical solution of Ito stochastic differential equations under the method of approximating multiple Ito and Stratonovich stochastic integrals based on generalized multiple Fourier series. Using the multiple stochastic integrals of multiplicity 1–3 appearing in the Ito–Taylor expansion as an example, it is shown that their expansions obtained using multiple Fourier–Legendre series are significantly simpler and less computationally costly than their analogs obtained on the basis of multiple trigonometric Fourier series. The results obtained in this paper can be useful for constructing and implementing strong numerical methods for solving Ito stochastic differential equations with multidimensional nonlinear noise.

Single Snapshot Super-Resolution DOA Estimation for Arbitrary Array Geometries

Abstract: We address the problem of search-free direction of arrival (DOA) estimation for sensor arrays of arbitrary geometry under the challenging conditions of a single snapshot and coherent sources. We extend a method of search-free super-resolution beamforming, originally applicable only for uniform linear arrays, to arrays of arbitrary geometry. The infinite dimensional primal atomic norm minimization problem in continuous angle domain is converted to a dual problem. By exploiting periodicity, the dual function is then represented with a trigonometric polynomial using a truncated Fourier series. A linear rule of thumb is derived for selecting the minimum number of Fourier coefficients required for accurate polynomial representation, based on the distance of the farthest sensor from a reference point. The dual problem is then expressed as a semidefinite program and solved efficiently. Finally, the search-free DOA estimates are obtained through polynomial rooting and source amplitudes are recovered through least squares. Simulations using circular and random planar arrays show perfect DOA estimation in noise-free cases.

Methodological Considerations on the Use of Different Spectral Decomposition Algorithms to Study Hippocampal Rhythms.

Abstract: Local field potential (LFP) oscillations are primarily shaped by the superposition of postsynaptic currents. Hippocampal LFP oscillations in the 25- to 50-Hz range (“slow γ”) are proposed to support memory retrieval independent of other frequencies. However, θ harmonics extend up to 48 Hz, necessitating a study to determine whether these oscillations are fundamentally the same. We compared the spectral analysis methods of wavelet, ensemble empirical-mode decomposition (EEMD), and Fourier transform. EEMD, as previously applied, failed to account for the θ harmonics. Depending on analytical parameters selected, wavelet may convolve over high-order θ harmonics due to the variable time-frequency atoms, creating the appearance of a broad 25- to 50-Hz rhythm. As an illustration of this issue, wavelet and EEMD depicted slow γ in a synthetic dataset that only contained θ and its harmonics. Oscillatory transience cannot explain the difference in approaches as Fourier decomposition identifies ripples triggered to epochs of high-power, 120- to 250-Hz events. When Fourier is applied to high power, 25- to 50-Hz events, only θ harmonics are resolved. This analysis challenges the identification of the slow γ rhythm as a unique fundamental hippocampal oscillation. While there may be instances in which slow γ is present in the rat hippocampus, the analysis presented here shows that unless care is exerted in the application of EEMD and wavelet techniques, the results may be misleading, in this case misrepresenting θ harmonics. Moreover, it is necessary to reconsider the characteristics that define a fundamental hippocampal oscillation as well as theories based on multiple independent γ bands.

Bernstein Fractal Trigonometric Approximation

Abstract: Fractal interpolation and approximation received a lot of attention in the last thirty years. The main aim of the current article is to study a fractal trigonometric approximants which converge to the given continuous function even if the magnitude of the scaling factors does not approach zero. In this paper, we first introduce a new class of fractal approximants, namely, Bernstein $\alpha $ -fractal functions using the theory of fractal approximation and Bernstein polynomial. Using the proposed class of fractal approximants and imposing no condition on corresponding scaling factors, we establish that the set of Bernstein $\alpha $ -fractal trigonometric functions is fundamental in the space of continuous periodic functions. Fractal version of Gauss formula of trigonometric interpolation is obtained by means of Bernstein trigonometric fractal polynomials. We study the Bernstein fractal Fourier series of a continuous periodic function $f$ defined on $[-l,l]$ . The Bernstein fractal Fourier series converges to $f$ even if the magnitude of the scaling factors does not approach zero. Existence of the $\mathcal{C}^{r}$ -Bernstein fractal functions is investigated, and Bernstein cubic spline fractal interpolation functions are proposed based on the theory of $\mathcal{C}^{r}$ -Bernstein fractal functions.

A semi analytical solution for free vibration analysis of combined spherical and cylindrical shells with non-uniform thickness based on Ritz method

Abstract: The paper investigated the free vibration of combined spherical-cylindrical-spherical (CSCS) shell with non-uniform thickness based on Ritz method. The energy method and first-order shear deformation theory are adopted to derive the formulas. The displacement functions are improved on base of domain decomposition method, in which the unified Jacobi polynomials are introduced to represent the displacement functions component along axial direction. The displacement functions component along circumferential direction is still Fourier series. In addition, the spring stiffness method is formed to be a unified format to deal with general boundary conditions. Then the final solutions are obtained by using Ritz method. To prove the validity of this method, the results of the same condition are compared with those obtained by FEM, published literatures and experiment. In addition, some meaningful examples are provided to reveal the free vibration characteristics of stepped spherical-cylindrical-spherical shell. The results of this paper can be used as the reference data for future research in related field.

Highly nonlinear wind waves in Currituck Sound: dense breather turbulence in random ocean waves

Abstract: We analyze surface wave data taken in Currituck Sound, North Carolina, during a storm on 4 February 2002. Our focus is on the application of nonlinear Fourier analysis (NLFA) methods (Osborne 2010) to analyze the data set: The approach spectrally decomposes a nonlinear wave field into sine waves, Stokes waves, and phase-locked Stokes waves otherwise known as breather trains. Breathers are nonlinear beats, or packets which “breathe” up and down smoothly over cycle times of minutes to hours. The maximum amplitudes of the packets during the cycle have a largest central wave whose properties are often associated with the study of “rogue waves.” The mathematical physics of the nonlinear Schrodinger (NLS) equation is assumed and the methods of algebraic geometry are applied to give the nonlinear spectral representation. The distinguishing characteristic of the NLFA method is its ability to spectrally decompose a time series into its nonlinear coherent structures (Stokes waves and breathers) rather than just sine waves. This is done by the implementation of multidimensional, quasi-periodic Fourier series, rather than ordinary Fourier series. To determine preliminary estimates of nonlinearity, we use the significant wave height Hs, the peak period Tp, and the length of the time series T. The time series analyzed here have 8192 points and T =1677.72 s = 27.96 min. Near the peak of the storm, we find Hs ≈ 0.55 m, Tp ≈ 2.4 s so that for the wave steepness of a near Gaussian process, ${S} = \left (\pi ^{5/2}/g\right )H_{s}/{T}_{p}^{2}$ , we find S ≈ 0.17, quite high for ocean waves. Likewise, we estimate the Benjamin-Feir (BF) parameter for a near Gaussian process, ${I_{BF}} = \left (\pi ^{5/2}/g \right ) H_{s} T/{T}_{p}^{3}$ , and we find IBF ≈ 119. Since the BF parameter describes the nonlinear behavior of the modulational instability, leading to the formation of breather packets in a measured wave train, we find the IBF for these storm waves to be a surprisingly high number. This is because IBF, as derived here, roughly estimates the number of breather trains in a near Gaussian time series. The BF parameter suggests that there are roughly 119 breather trains in a time series of length 28 min near the peak of the storm, meaning that we would have average breather packets of about 14 s each with about 5-6 waves in each packet. Can these surprising results, estimated from simple parameters, be true from the point of view of the complex nonlinear wave dynamics of the BF instability and the NLS equation? We analyze the data set with the NLFA to verify, from a nonlinear spectral point of view, the presence of large numbers of breather trains and we determine many of their properties, including the rise time for the breathers to grow to their maximum amplitudes from a quiescent initial state. Energetically, about 95% of the NLFA components are found to consist of breather trains; the remaining small amplitude components are sine and Stokes waves. The presence of a large number of densely packed breather trains suggests an interpretation of the data in terms of breather turbulence, highly nonlinear integrable turbulence theoretically predicted for the NLS equation, providing an interesting paradigm for the nonlinear wave motion, in contrast to the random phase Gaussian approximation often considered in the analysis of data.

Tensor calculus in spherical coordinates using Jacobi polynomials. Part-I: Mathematical analysis and derivations

Abstract: This paper presents a method for accurate and efficient computations on scalar, vector and tensor fields in three-dimensional spherical polar coordinates. The method uses spin-weighted spherical harmonics in the angular directions and rescaled Jacobi polynomials in the radial direction. For the 2-sphere, spin-weighted harmonics allow for automating calculations in a fashion as similar to Fourier series as possible. Derivative operators act as wavenumber multiplication on a set of spectral coefficients. After transforming the angular directions, a set of orthogonal tensor rotations put the radially dependent spectral coefficients into individual spaces each obeying a particular regularity condition at the origin. These regularity spaces have remarkably simple properties under standard vector-calculus operations, such as gradient and divergence. We use a hierarchy of rescaled Jacobi polynomials for a basis on these regularity spaces. It is possible to select the Jacobi-polynomial parameters such that all relevant operators act in a minimally banded way. Altogether, the geometric structure allows for the accurate and efficient solution of general partial differential equations in the unit ball.